Further Discussions on Induced Bias Matrix Model for the Pair-wise Comparison Matrix

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Further Discussions on Induced Bias Matrix Model for the Pair-wise Comparison Matrix

^

Daji Ergu^1,2 Gang Kou¹ Yi Peng^1,* János Fülöp³ Yong Shi^4,5

Abstract Inconsistency issue of pairwise comparison matrices has been an important subject in the analytical network process. The most inconsistent elements can efficiently be identified by inducing a bias matrix only based on the original matrix.

This paper further discusses the induced bias matrix, and integrates all related theorems and corollaries into the induced bias matrix mode. The theorem of inconsistency identification is proved mathematically using the maximum eigenvalue method and the contradiction method. In addition, a fast inconsistency identification method for one pair of inconsistent elements is proposed and proved mathematically.

Two examples are used to illustrate the proposed fast identification method. The results show that the proposed new method is easier and faster than the existing method for the special case with only one pair of inconsistent elements in the original comparison matrix.

AMS Classification Number: 47N10; 65K10.

This research has been partially supported by grants from the National Natural Science Foundation of China (#70901011 and #71173028 for Yi Peng, #70901015 for Gang Kou, and #70921061 for Yong Shi), the Fundamental Research Funds for the Central Universities and Program for New Century Excellent Talents in University (NCET-10-0293).

1 School of Management and Economics, University of Electronic Science and Technology of China, Chengdu, 610200, China.

2 Southwest University for Nationalities, Chengdu, 610054, China.

3 Research Group of Operations Research and Decision Systems, Computer and Automation Research Institute, Hungarian Academy of Sciences, H-1111 Kende u. 13-17, Budapest, Hungary.

4 College of Information Science & Technology, University of Nebraska at Omaha, Omaha, NE 68182, USA.

5 Research Center on Fictitious Economy and Data Sciences, Chinese Academy of Sciences, Beijing 100190, China.

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Keywords： Analytic network process (ANP); the induced bias matrix model (IBMM);

Inconsistency identification; Reciprocal pairwise comparison matrix (RPCM)

1. Introduction

The pair-wise comparison method is a well-established technique, and widely used in multi-criteria decision making (MCDM) methods [1, 2]. The consistency test of pair-wise comparison matrix in the AHP and ANP, two of the widely used MCDM methods, has been studied extensively over the past few decades [3-14]. To improve the consistency ratio of the pair-wise comparison matrix in the AHP and ANP and preserve the original comparison information as much as possible, literature [15]

introduced an induced bias matrix (IBM, hereinafter), which can be derived by the original reciprocal pairwise comparison matrix (RPCM hereinafter), to identify and adjust the inconsistent elements.

IBM is based on the theorems of matrix multiplication and vectors dot product as well as the definitions and notations of the pair-wise comparison matrix. The IBM method has been applied in questionnaire design [16], risk analysis [17], and task scheduling [18]. If the comparison matrix A is perfectly consistent, we mathematically proved that the IBM should be a zero matrix in [15]. If the comparison matrix A is approximately consistent, we also mathematically proved that the IBM should be as close as possible to a zero matrix in [19]. This corollary can be used to estimate the uncertain or missing values in an RPCM. If the pair-wise matrix A is inconsistent, there must be some inconsistent elements in the induced bias matrix (IBM) deviating far away from zero (Corollary 2 in [15]). This corollary shows that

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the farthest value should be identified as the most inconsistent element from the induced bias matrix (IBM). However, this critical corollary for identifying the most inconsistent element has not been proved mathematically in [15].

The objective of this paper is to prove the theorem of aforementioned critical corollary mathematically, and integrates all related theorems and corollaries into the induced bias matrix model (IBMM), which simplifies and refines the proposed inconsistency identification method. In addition, this paper proposes a fast inconsistency identification method to extend the proposed IBMM for the special case of one pair of inconsistent elements in the original RPCM.

The remaining parts of this paper are organized as follows. The next section integrates all related theorem and corollaries into one model and provides two mathematic proofs of Corollary 2.2 (i.e. Theorem 2.3 in IBMM) by maximum eigenvalue method and contradiction method. Section 3 analyzes the inconsistency identification method; proposes and proves a fast inconsistency identification method;

describes the sign of non-zero of the induced bias matrix; and introduces two numeric examples with one pair of inconsistent elements to demonstrate the proposed fast inconsistency identification method. Section 4 concludes the paper.

2. Theorems and Proofs of the Induced Bias Matrix Model (IBMM)

In order to efficiently identify the inconsistent elements and preserve most of the original pair-wise comparison information, we proposed an induced bias matrix (IBM), which is only based on the original RPCM in [15], and the following theorem and corollaries were derived.

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Theorem 2.1: The induced bias matrix (IBM) CAAnA should be a zero matrix, if comparison matrix A is perfectly consistent.

Corollary 2.1: The induced bias matrix (IBM)

C  AA  nA

should be as close as possible to zero matrix, if comparison matrix A is approximately consistent.

Corollary 2.2: There must be some inconsistent elements in induced bias matrix (IBM)

C

deviating far away from zero, if the pair-wise matrix A is inconsistent.

The correctnesses of Theorem 2.1 and Corollary 2.1 have been proved mathematically in Section 3.1 of [15] and in Section 2.1 of [19], respectively. Besides, some special cases, where there are some errors in the original RPCM of order 3, have been addressed as examples in [19]. To determine whether a comparison pairwise matrix is reciprocal, Corollary 2.3 is proposed and illustrated using 33 comparison pair-wise matrix, as an example in [19].

Corollary 2.3:Despite that the comparison matrix A is consistent or not, all entries in the main diagonal of the induced bias matrix (IBM) C AAnA should be zeroes, giving that the comparison matrix A is satisfied with the reciprocal condition.

The inconsistent elements are identified by inducing a bias matrix C from the original RPCM, and the critical component of the above mentioned theorem and corollaries is the induced bias matrix (IBM). To simplify and refine the proposed inconsistency identification method, and make the proposed method more comprehensive and systematic, we integrate these theorem and corollaries into one

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model, say induced bias matrix model (IBMM), which includes the following three theorems.

The Theorem of the Induced Bias Matrix Model (IBMM):

Theorem 2.2: The induced bias matrix (IBM) CAAnA should be equal (or close) to a zero matrix, if comparison matrix A is perfectly (or approximately) consistent. That is,











 

 ，

，

ij kj ik

a a a if

a a a if nA

AA

C 0

0

(1)

where A is the original RPCM and a_ij represents the values of RPCM. The “n” denotes the order of RPCM.

Theorem 2.3:There must be some inconsistent elements in the induced bias matrix (IBM)

C

deviating far away from zero, if the pair-wise matrix is inconsistent.

Especially, any row or column of matrix C contains at least one positive element.

Theorem 2.4: All entries in the main diagonal of the induced bias matrix (IBM) nA

AA

C  should be zeroes whether matrix A is consistent or not, as long as the comparison matrix A satisfies the reciprocal condition.

We provided the proof of Theorem 2.2 for consistent case in [15]. The principles of Theorem 2.3 and Theorem 2.4 are demonstrated by introducing some errors into a

3

3 RPCM in [19]. The following subsections mathematically prove Theorem 2.3 and Theorem 2.4.

2.1 The Proof of Theorem 2.3 by Maximum Eigenvalue Method

Proof: If the RPCM A is inconsistent, the induced bias matrix C AAnA

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cannot be zero. More precisely, any row of C contains at least one positive element.

It is known, e.g. literature [20], that for the maximal eigenvalue _max of A,

n

max , and the corresponding unique eigenvector _max is a positive vector.

Furthermore, A is consistent if and only if _max n. By applying

A_{m a x}_{m a x}_{m a x} (2) at the appropriate places, we get

 



_max



_max.

max max

2 max

max max max























n n

n A

nA AA C















 (3)

Since _max >n, C_max is a positive vector. Consequently, C cannot have any row containing only zeros. Moreover, since both C_max and _max are positive vectors, any row of C must contain at least one positive element. □

2.2 The Proof of Theorem 2.3 by Contradiction

Proof: It has been proved in [15] that, if a reciprocal pairwise comparison matrix (RPCM) is perfectly consistent, that is, a_ij a_ika_kj for all i,j,k, then

. 0

1

















ij ij ij n

k

kj ik

ij a a na na na

c (4)

If an RPCM A is inconsistent, a_ij a_ika_kj holds at least for one of the i,j,k



i,j,k1,2,,n



. Moreover, if A is inconsistent, for any i, there exist j and k such that a_ij a_ika_kj (Corollary 2 in [21]). Assume that an RPCM A is inconsistent, but the i-th row of the induced bias matrix C contains only non-positive elements. Then

kj ik

ij a a

a  with some j and k, and c_i₁ 0,c_i₂ 0,,c_in 0. We get the following inequalities:

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































, 0

0 0

1

2 1

2 2

1 1

in n

k

kn ik in

i n

k k ik i

i n

k k ik i

na a a c



(5)



















1 . 1 1

1 1

2 2

1 1 1

n a a a

n

k

kn ik in

n

k k ik i

n

k k ik i



(6)

Adding all the inequalities together in the system of inequalities (6), we get

2 1

1 2 1 2

1 1

1 a a n

a a a a

a a a

n

k

kn ik in n

k k ik i

n

k k ik i





 



  

 , (7)

 1 1 1 ₂,

1 1

2 1 2

1 1

n a a a a

a a a

a a

n

k

kn ik in n

k

k ik i n

k

k ik i





 



  

 (8)

 1 2.

1 1

a n a a a

a a

n

j n

k ij

ik kj n

j n

k

kj ik ij







   

(9)

The inequality (8) or (9) can be unfolded to the following matrix form:

1 . 1

1 1

2 3

3 2

2 1

1

3 3 2

2 1

1

3 3 2

2 1

1

3 3 3

3 3

3 33

3 3 23 2 3 13 1 3

2 2 2

2 2

2 32

3 2 22 2 2 12 1 2

1 1 1

1 1

1 31

3 1 21 2 1 11 1 1

n a a a a

a a a

a a a a a a a a

a a a a

a a a

a a a a a a a a

a a a a

a a a

a a a a a a a a

a a a a

a a a

a a a a a a a a

a a a a

a a a

a a a a a a a a

a a a a

a a a

a a a a a a a a

nn in in jn

ij in in

ii in n

i in n i in n i in

nj in ij jj

ij ij ij

ii ij j

i ij j i ij j i ij

ni in ii ji

ij ii ii

ii ii i

i ii i i ii i i ii

n in i j

ij i i

ii i i

i i

i i i

n in i j

ij i i

ii i i

i i i i i

n in i j

ij i i

ii i i

i i i i i



















1 . 1

1 1

2 3

3 2

2 1

1

3 3 2

2 1

1

3 3 2

2 1

1

3 3 3

3 3

3 33

3 3 23 2 3 13 1 3

2 2 2

2 2

2 32

3 2 22 2 2 12 1 2

1 1 1

1 1

1 31

3 1 21 2 1 11 1 1

n a a a a

a a a

a a a a a a a a

a a a a

a a a

a a a a a a a a

a a a a

a a a

a a a a a a a a

a a a a

a a a

a a a a a a a a

a a a a

a a a

a a a a a a a a

a a a a

a a a

a a a a a a a a

nn in in jn

ij in in

ii in n

i in n i in n i in

nj in ij jj

ij ij ij

ii ij j

i ij j i ij j i ij

ni in ii ji

ij ii ii

ii ii i

i ii i i ii i i ii

n in i j

ij i i

ii i i

i i

i i i

n in i j

ij i i

ii i i

i i i i i

n in i j

ij i i

ii i i

i i i i i



















(10)

(8)

Since matrix A is a reciprocal matrix, that is,

jk

kj a

a  1 and a_kj 0, a_ij 0,

0

aik , from the expansion inequality (9), any of the above inequalities (7)-(10) can be simplified as the following inequality:

n²,

a a a a a a n

j

k ik

ij jk ij ik

kj 









 







(11)

 n² n.

a a a a a a

j

k ik

ij jk ij ik

kj  









 





(12)

Since there are 2

) 1 (n

n sum term at the left side of the inequality (12), and

1 2







ij ik ij kj

ik kj ik ij jk ij ik kj

a a a a a a a a a a

a a , the inequality (12) holds if and only if

1

ij ik kj a

a a , namely, a_ij a_ika_jk for all j and k. However, this result contradicts the

previous assumption that a_ij a_ika_kj for some j and k. Therefore, one of the inequalities, at least, does not hold. Thus, inequality (12) holds with > sign. This entails that at least one of elements in the i-th row the induced bias matrix C is

positive. □

Based on the above two proofs for rows, the same proofs for columns can also be induced. If A is a pairwise comparison matrix with the reciprocal property, the transpose of A is also a pairwise comparison matrix with the reciprocal property. In addition, A is consistent if and only if the transpose of A is consistent.

The transpose of the IBM C generated by A is the IBM generated by the transpose of A. Consequently, if C is inconsistent, any column of C contains at least one positive element. The same statement for the rows was stated earlier

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2.3 The Proof of Theorem 2.4

Proof: According to the principle of matrix multiplication, all values in the main diagonal of the induced bias matrix C can be calculated by the formula (13):

.

1

ii n

k

ki ik

ii a a na

c 



 



(13)

If 1 , 1



 _ii

ik

ik and a

a a , then

. , , 2 , 1 , 1 0

1 1

n i

n n a na

a na

a a

c _ii

n

k ik

ik ii

n

k

ki ik

ii 



  



      



(14)

□

3. The Inconsistency Identification Method

In this section, the mathematic principles of the proposed “Method of Maximum”,

“Method of Minimum” and “Method for identifying

a

_ij” in [15], are firstly discussed. Next, a fast inconsistency identification method for special case is proposed and proved mathematically. Two numerical examples are introduced to illustrate the proposed method in Section 3.3. Details are given next.

3.1 Clarification of the Proposed Inconsistency Identification Method

If the RPCM A is inconsistent, the above proof shows that any row of the IBM C contains at least one non-zero element, that is, a_ij a_ika_kj holds at least for one group of i,j,k, which means that there is at least one pair of inconsistent elements existing in the original RPCM A. Suppose c_ij, the element with largest absolute value in the IBM C, is identified. The second step is to analyze that which element makes cij to be far away from zero. According to the rule of matrix multiplication, the value

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of c_ij is calculated by all values on the

i

^th row and

j

^th column of matrix A and aij, that is,

_ij

n

k

kj ik

ij a a na

c 



 

1

(15)

2 .

2 1

1 j ij i j ij ik kj ij in nj ij

i a a a a a a a a a a a

a         

  

Clearly, the farthest value of c_ij can be impacted by any term of a_ika_kja_ij on the right side of the sum equality (15). In order to identify the inconsistent elements that caused the value of c_ij to be far away from zero, the scalar product of vectors in n dimension technique is introduced. The impact of each term can easily be observed by the scalar product of vectors in n dimension technique, that is,



i i in

 

j j nj



T j

i c a a a a a a

r

b   ₁, ₂,,  ₁ , ₂ ,, 



ai₁a₁j,ai₂a₂j,,ainanj



^， (16) and





b a_ij

f



ai1a1j aij^,ai2a2j aij^,aikakj aij^,^,ainanjaij



^. (17) If a_ij a_ika_kj, then a_ika_kj a_ij 0. Therefore, the non-zero element(s), which caused the value of c _ij to be far away from zero, can be identified through observing all elements in the bias identifying vector

f

. In addition, the inequality a_ij a_ika_kj can be caused by a_ij or any a_ika_kj



^k¹^,²^,^^,ⁿ



, or both.

Obviously, if the inconsistent element is a_ij, other elements are consistent.

Assume a_ika_kj a_ij a_ij, namely, a _ij is too small. We can get that all values in the bias identifying vector

f

are positive except ki,j , as a_iia_ij a_ij 0 and

0

 _ij

jj

ija a

a . Vice versa, if a _ij is too large, all the values in the bias identifying vector

f

will be negative except two values k i,j. Therefore, the “Method for identifyinga_ij” inconsistency identification method is proposed [15].

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Besides, the farthest value of c_ij must be caused by some outliers either too large or too small located at the bias identifying vector

f

; therefore, “Method for Maximum” and “Method for Minimum” inconsistency identification methods are proposed [15].

In order to further identify the inconsistent element for those elements whose values are close to the largest or smallest simultaneously, therefore the “Method of matrix order reduction” inconsistency identification method is proposed [15].

3.2 Fast Identification Method for Special Case and Its Proof

In this section, one fast inconsistency identification method is proposed to quickly identify the inconsistent elements when there is only one pair of inconsistent elements in the original RPCM.

Assume that RPCM A is inconsistent, and there is one pair of inconsistent elements a_ip and its corresponding reciprocal element

ip

pi a

a  1 , while other elements are consistent, namely, a_ika_kj a_ij for all k except k  p (a_ipa_pj a_ij).

Therefore, the two inconsistent elements are elements at the i^th and p^th rows, and the p^th and i^th columns. According to the rule of matrix multiplication, all elements, which are located at the i^th, p^th rows, and the i^th,p^th columns in the induced bias matrix C AAnA, will be impacted by a_ip and a_pi. Since it is assumed that a_ika_kj a_ij



k p;j p,



and a_ika_kp a_ip, suppose a_ika_kp a_ip , all the values in the i^th row of the IBM C can be computed by formula (18), that is,

c a a na a a a_ipa_pj na_ij j n

n

p k

kj ik ij

n

k

kj ik

ij , 1,2, ,

, 1 1

 















 





(12)

 

     









 









 

























.

; 2

2

; 0 1

1

,

; 1

p j a a n na a

a a a a n

i j n

n

p i j a a a na a a a n

ip ip ip

pp ip ip ii ip

ij pj ip ij pj ip ij

(18)

In order to analyze the sign change of each element on the i^th row, the equalities in (18) are further unfolded, as shown below.

   



















 









 













.

) (

2 1

0

2 2 2

1 1 1

in pn ip in

ip ip ip

pp ip ip ii ip ip

ii

i p ip i

a a a c

a a n na a a a a a n c c

a a a c



(19)

If a_ip , then

   























 









 

















. 0

0 ) (

2 1

0

0 0

2 2 2

1 1 1

in pn ip in

ip ip ip

pp ip ip ii ip ip

ii

i p ip i

a a a c



(20)

If a_ip , then

   























 









 

















, 0

0 ) (

2 1

0

0 0

2 2 2

1 1 1

in pn ip in

ip ip ip

pp ip ip ii ip ip

ii

i p ip i

a a a c



(21)

where the symbols “” and “” denote “increase” and “decrease”, respectively

(13)

(hereinafter).

Likewise, for the p^th row:

n j

na a a a a na

a a

c _pi _ij _pj

n

i k

kj pk pj

n

k

kj pk

pj , 1,2, ,

, 1 1

 















 





 

     









 









 



















.

; 2

1

; 0

,

; 1

i j a a n na a a a a a n

p j

p i j a a a na a a a n

pi pi pi

pi pp ii pi pj

pj ij pi pj ij pi pj

(22)

Therefore, if a_ip , then  

ip

pi a

a 1

,

 

     











 









 





















.

; 0 2

1

; 0

,

; 0 1

i j a

a n na a a a a a n

p j

p i j a

a a na a a a n c

pi pi pi

pi pp ii pi pj

pj (23)

Likewise, if a_ip , then  

ip

pi a

a 1

,

 

     











 









 





















.

; 0 2

1

; 0

,

; 0 1

i j a

a n na a a a a a n

p j

p i j a

a a na a a a n c

pi pi pi

pi pp ii pi pj

pj (24)

If a_ip , all values on the i^th row of IBM C will be more than zeroes (c_ij 0,j1,2,,n and j p) except c_ip 0, and all values on the p^th row of IBM C will be less than zeroes (c_ij 0,j1,2,,n and j p) except c_ip 0. Therefore, only the elements on the i^th row and p^th row are non-zeroes, and the sign form of the values on the i^th row and p^th row of the IBM C can be derived, as shown in the following matrix,

(14)

. 0

0

th th th

th

p i p

i































 (25)

Likewise, the signs of each element on the i^th column and p^th column can be derived similarly.

n j

na a a a a na

a a

c _jp _pi _ji

n

p k

ki jk ji

n

k

ki jk

ji , 1,2, ,

, 1 1

 















 





 

     









 









 



















.

; 2

1

; 0

,

; 1

p j a a n na a a a a a n

i j

i p j a a a na a a a n

pi pi pi

pi pp ii pi pi

ji pi jp ji pi jp ji

(26)

If a_ip , then  

ip

pi a

a 1

,

 

     











 









 





















.

; 0 2

1

; 0

,

; 0 1

p j a

a n na a a a a a n

i j

i p j a

a a na a a a n c

pi pi pi

pi pp ii pi pi

ji pi jp ji pi jp ji

ji (27)

For the p^th column,

n j

na a a a a na

a a

c _ji _ip _jp

n

i k

kp jk jp

n

k

kp jk

jp , 1,2, ,

, 1 1

 















 





 

     









 









 



















,

; 2

1

; 0

,

; 1

i j a a n na a a a a a n

p j

i p j a a a na a a a n

ip ip ip

pp ip ip ii ip

jp ip ji jp ip ji jp

(28)

If a_ip ,

 

     











 









 





















.

; 0 2

1

; 0

,

; 0 1

i j a

a n na a a a a a n

p j

i p j a

a a na a a a n c

ip ip ip

pp ip ip ii ip

jp ip ji jp ip ji jp

jp (29)

Therefore, the sign forms of the elements on the i^th and p^th columns of the